There have never been any migration records between England/Wales and Ireland. It was all one country and you were just moving from one county to another. Very few parishes records go back to the 1700s and so in the majority of cases it is very difficult or impossible to find details of marriages and births for the 1600s and 1700s in Ireland. I think your problem is not that the information you are looking for is restricted or private, just that it doesn’t exist any more.
You don’t say what denomination the family were but if they were Church of Ireland (ie Anglican), then copies of the surviving parish records are kept at the Representative Church Body library in Dublin. Be aware however that a lot of old COI records were destroyed in the 1922 civil war, having ironically been sent to Dublin for safe keeping.